On transitivity of parametric family of cardinality-based fuzzy similarity measures

Abstract

In crisp logic every object is similar to itself with degree of reflexivity 1, while the degree of reflexivity in fuzzy logic can be any value in the unit interval, [0, 1]. This behaviour of a fuzzy set is used to enlighten the concept of similarity and inclusion measures. This paper is aimed at discovering the relations between the parameters of the transitive members of a family of cardinality-based fuzzy measure.

Keywords

Cardinality based similarity measures fuzzy reflexive measures

1 Introduction

People are often facing the situations in which they prefer one object over the other or they compare them in their daily life on the basis of features or attributes. This process is commonly known as similarity and inclusion measures. Most of the similarity measures are originated from Taxonomy [28], where feature-based approach is followed. While studying these measures (similarity and inclusion) Mathematically, the objects are normally represented by the sets of features. For example, when comparing two cars these sets may contain their colors, safety ratings and their performances. Therefore, similarity and inclusion measures play a vital role in organizing and classifying the objects, plants, species, etc. [18, 24] and in the association between the species [13, 25]. It is a very important concept in many scientific fields such as Bioinformatics, Biology, Chemistry, Information retrieval, Statistics and many others [3, 7, 17, 27, 29, 30] and [32].

Similarity measure determines the degree of resemblance between two objects while the inclusion measure expresses the degree to which the characteristics of the one object include another. These measures received much more attention in recent decades because of their application in decision making, pattern recognition, medical diagnosis and data mining applications.

Lotfi A. Zadeh introduced the notion of fuzzy sets [35] in 1965, which have an edge over the crisp sets that they can represent the degrees of truth or falsehood only. Fuzzy set define the intermediate values between the complete truth and complete false and hence fuzzy measures. Consequently, in many practical applications, fuzzy similarity measures which represent the degree of similarity between two objects seem much closer to reality than their crisp counterparts do. Since their inception in 1973, fuzzy similarity and inclusion measures found almost all application areas from A to Z (i.e., from Anthropology to Zoology). Cardinality based similarity and inclusion measures play a key role in literature and applications [1 , 15] and [34]. Some memorable contributions can be found in [2, 8–10 , 20] and [21].

Cardinality based similarity and inclusion measures rely on the cardinalities of sets of common and different features. New measures were introduced by assigning different weights to the sets of common and different features. Such families of similarity and inclusion measures are known as parametric families and these weights are termed as parameters. Some famous parametric families were introduced by Tversky [30], Gower & Legendre [16] and De Baets et al. [9]. A family of rational expressions using two, three and four parameters [4 , 12] and [20] are available in literature. While axioms for fuzzy similarity measures are focused, fuzzy equivalence relation stands as the best model for fuzzy similarity measure. A fuzzy equivalence relation is reflexive, symmetric and T-transitive [31] fuzzy relation. Unfortunately, the majority of fuzzy similarity measures found in literature do not follow one or the other desired form of fuzzy transitivity, in particular, the axiom of T-transitivity is violated. In [20] Janssens et al. established conditions under which a parametric family of cardinality based fuzzy similarity measures become T-transitive. The significant development of meta-theorems, which ensure T-transitivity, can be found in [11] and [21]. These theorems are used to construct the necessary and sufficient conditions to obtain transitivity.

De Baets et al introduced a parametric family of cardinality based similarity measure [8] and inclusion measure [9], which cover most of the famous measures found in literature for different values of parameters. Basically, these measures used eight parameters and Janssens et al. characterized the T-transitive members of the family of similarity and inclusion measures for only four parameters for Łukasiewicz, Product and Min t-norms.

Sometimes in similarity or inclusion measure, the features which are absent or negative match play a key role in the comparison of two objects and it seems improper to ignore these negative match features. In this context, a manuscript is already been submitted by Javed, Samina and Syed using six parameters to characterize the Łukasiewicz-transitive members and fuzzy reflexive measures using Łukasiewicz t-norm is accepted [19] while this submission not only tried to explore the transitivity between the objects based on the similarities but also to incorporate the negative matching features. This paper is focused to determine the necessary conditions while extending the number of parameters to eight for Łukasiewicz-, Product- and Min-transitive family of similarity and inclusion measures based on cardinality.

By using four parameters, the family introduced by De Baets et al covers some of the inclusion and similarity measures available in literature [9, 33] but fails to define some measures like Kuncheva [23]. This shortcoming can be cover by extending the number of parameter. So, it is an effort to include some more measures in this parametric family, which were put out of action by the restriction of four parameters.

The notion of fuzzy set was introduced by Zadeh in 1965 in his seminal paper [35]. A fuzzy set A is a mapping from a universe X to [0, 1]. For any x ∈ X, the value A (x) denotes the degree of membership of x in A. Let F (X) be the set of all fuzzy subsets of a universe X. For a crisp universe X, a fuzzy subset of X × X is called a fuzzy binary relation and throughout this paper we termed fuzzy binary relations as fuzzy relations. Given a crisp universe X, and A, B ∈ F (X), A is said to be a subset of B (in Zadeh’s sense [35]) denoted by A ⊆ B, if and only if A (x) ≤ B (x) for all x ∈ X.

Definition 1.1. [26] The triangular norm (t-norm) T and triangular conorm (t-conorm) T^∗ are increasing, associative, commutative and mapping [0, 1] ² → [0, 1] satisfying T (1, x) = x and T^∗ (x, 0) = x for all x ∈ [0, 1].

To every t-norm T there corresponds a t-conorm T^∗ called the dual t-conorm, defined by: T^∗ (x, y) =1 - T (1 - x, 1 - y) . For the Lukasiewicz t-norm W (x, y) = max(x + y - 1, 0) , the corresponding t-conorm is W^∗ (x,) = min(x + y, 1) .

Definition 1.2. [14] A negator N is an order-reversing [0, 1] → [0, 1] mapping such that N (0) =1 and N (1) =0. A strictly decreasing negator satisfying N (N (x)) = x for all x ∈ [0, 1] is called a strong negator.

The negator defined as: N (x) =1 - x for all x ∈ X, is called standard negator and was defined by Zadeh himself.

Definition 1.3. [31] Given a t-norm T, a T-equivalence relation on a set X is a fuzzy relation E on X that satisfies:

E (x, x) =1 for all x ∈ X; (Reflexivity),

E (x, y) = E (y, x) for all x, y ∈ X; (Symmetry),

T (E (x, y) , E (y, z)) ≤ E (x, z) for all x, y, z ∈ X; (T-transitivity).

Definition 1.4. [14] Let R be a fuzzy relation on X . A triangular norm T is T-transitive if and only if for all x, y, z ∈ X, T (R (x, y) , R (y, z)) ≤ R (x, z) .

The Lukasiewicz T_L- and Product T_P-transitive similarity measures are of special interests. A similarity measure S is Lukasiewicz-transitive if it satisfies $S (A, B) + S (B, C) - 1 \leq S (A, C)$ (1) for all A, B, C ∈ P (X). A similarity measure S is Product-transitive if it holds $S (A, B) \cdot S (B, C) \leq S (A, C)$ (2) for all A, B, C ∈ P (X). Similarly, a measure S is Min-transitive if it holds $\min (S (A, B), S (B, C)) \leq S (A, C)$ (3) for all A, B, C ∈ P (X).

Definition 1.5. [31] An inclusion measure for ordinary sets is binary fuzzy relation I on the power set P (X) = { 0, 1 } ^X satisfying, A ⊂ B ⇒ I (A, B) = 1.

2 Transitivity of inclusion measures

In 2001, De Baets et al. [8] introduced a rational cardinality based inclusion measure. This parametric family contains eight parameters to define inclusion measure for two subsets A and B in a finite universe X and measure the degree of inclusion of one set in the other set as $I (A, B) = \frac{x χ_{A, B} + t χ_{B, A} + y δ_{A, B} + z ν_{A, B}}{x^{'} χ_{A, B} + t^{'} χ_{B, A} + y^{'} δ_{A, B} + z^{'} ν_{A, B}}$ (4) with χ_A,B = |A ∖ B|, χ_B,A = |B ∖ A|, δ_A,B = |A ∩ B|, ν_A,B = | (A ∪ B) ^c|, etc. and the parameters x, x′, t, t′, y, y′, z, z′ ∈ {0, 1}. These parameters are considered to be positive real numbers. The conditions of x < x′ and t < t′ are imposed to contain inclusion measure I (A, B) in unit interval, as x = x′, t = t′ leads to the trivial case, so we consider 0 ≤ x < x′ and 0 ≤ t < t′.

Fig.1

Cardinalities associated with sets A, B and C.

Janssens et al. [20] restricted the number of parameters of the family to four parameters x, x′, y and z and explore the necessary and sufficient condition on the parameters for T-transitivity. In this section, we determine the conditions on parameters by extending their number up to eight parameters x, x′, t, t′, y, y′, z and z′ for T-transitivity. For this purpose, the notion of fuzzy reflexivity of a measure is used. The inclusion measure will be fuzzy reflexive if y′ = θy, z′ = φz for θ, φ ≥ 1, so the above inclusion measure can be written as $I (A, B) = \frac{x χ_{A, B} + t χ_{B, A} + y δ_{A, B} + z ν_{A, B}}{x^{'} χ_{A, B} + t^{'} χ_{B, A} + y θ δ_{A, B} + z φ ν_{A, B}}$ (5) with θ, φ ≥ 1. For six parameters, the above inclusion measure can be written as $I (A, B) = \frac{x χ_{A, B} + x^{'} χ_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} Δ_{A, B} + y θ δ_{A, B} + z ϕ ν_{A, B}}$ (6) where y′ = θy, z′ = φz for θ, φ ≥ 1 and ▵_A,B = |A ∖ B| + |B ∖ A|. Consider the setting in figure, then following conditions hold,

$\begin{matrix} | A ∖ B | = a_{1} + b_{2} \\ | B ∖ C | = a_{2} + b_{3} \\ | A ∖ C | = a_{1} + b_{3} \\ | A \cap B | = b_{3} + c \\ | A \cap C | = b_{2} + c \\ | B \cap C | = b_{1} + c \\ | B ∖ A | = a_{2} + b_{1} \\ | C ∖ B | = a_{3} + b_{2} \\ | C ∖ A | = a_{3} + b_{1} \\ | {(A \cup B)}^{c} | = a_{3} + d \\ | {(A \cup C)}^{c} | = a_{2} + d \\ | {(B \cup C)}^{c} | = a_{1} + d \\ | A Δ B | = a_{1} + b_{2} + a_{2} + b_{1} \\ | A Δ C | = a_{1} + b_{3} + a_{3} + b_{1} \\ | B Δ C | = a_{2} + b_{3} + a_{3} + b_{2} \end{matrix}$ (7)

where a_i′s, b_i′s, c and d are cardinalities. In this paper, Lukasiewicz, Product and Min t-norms are used to articulate the conditions for transitivity, which are termed as T_L - , T_P- and T_M-transitivity throughout the paper.

2.1 Lukasiewicz transitive members

Theorem 2.1.1. The T_L-transitive members of the class of inclusion measures (ref equ2.03) are characterized by the necessary condition $x^{'} \geq max (y θ, z φ) .$

Proof. The inclusion measure I is T_L-transitive if it fulfill the definition of transitivity (1). To determine the conditions on parameters x, x′, y, and z for transitivity, the inequality (1) can be written in terms of family (6) as

Using the values of cardinalities given in (7), the above inequality implies

Setting a₁ = b₁ = c = d = 0, (8) implies

$\begin{matrix} (x^{'} - x) [\begin{matrix} \frac{b_{2}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2})} \\ - \frac{b_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} \end{matrix}] \\ + y (θ - 1) [\begin{matrix} \frac{b_{3}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} \\ - \frac{b_{2}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} \end{matrix}] \\ + z (φ - 1) [\begin{matrix} \frac{a_{3}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} \\ - \frac{a_{2}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0 \end{matrix}$

Since x′ > x, θ ≥ 1 and φ ≥ 1, so the factors x′ - x, θ - 1 and φ - 1 can be omitted and we obtain

Expanding the above inequality and setting a₂ = a₃ = 0 leads to (x′) ² - (x′ - yθ) yθ ≥ 0 which can be satisfied when x′ ≥ yθ . Other combinations of a_i′s and b_i′s, that is, b₂ = b₃ = 0, a₂ = b₃ = 0, or b₂ = a₃ = 0 do not lead to any other conditions on x, x′, y, z. Setting a₂ = b₂ = c = d = 0 in (8). Since x′ > x, θ ≥ 1 and φ ≥ 1 so the factors x′ - x, θ - 1 and φ - 1 can be omitted and we obtain

$\begin{matrix} [\begin{matrix} \frac{a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1})} \\ + \frac{b_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{1} + z φ a_{1}} \end{matrix}] \geq 0 \\ [\begin{matrix} (x^{'} - y θ) (x^{'} b_{1}^{2} b_{3} + z φ a_{3} b_{1} b_{3} + y θ b_{1} b_{3}^{2}) \\ + (x^{'} - z φ) (x^{'} a_{1} a_{3}^{2} + y θ a_{1} a_{3} b_{1} + z φ a_{1}^{2} a_{3}) \\ + (2 {(x^{'})}^{2} - y^{2} θ^{2} - {zx}^{'} φ) a_{1} b_{1} b_{3} \\ + (2 {(x^{'})}^{2} - z^{2} φ^{2} - {yx}^{'} θ) a_{1} a_{3} b_{3} \\ + ({(x^{'})}^{2} - yz θ φ) (a_{1} b_{3}^{2} + a_{1}^{2} b_{3}) \end{matrix}] \geq 0 \end{matrix}$ (9)

In particular, setting a₁ = a₃ = 0 in the above inequality, it will be true if x′ ≥ yθ. Similarly, by setting b₁ = b₃ = 0, that is, x′ ≥ zφ is necessary to hold the inequality and setting of b₁ = a₃ = 0, ends up with a weaker condition (x′) ² - yzθφ ≥ 0 . Other combinations of a_i′s and b_i′s do not lead to any other conditions on x, x′, y, z. Setting a₃ = b₃ = c = d = 0 in (8). Since x′ > x, θ ≥ 1 and φ ≥ 1 so the factors x′ - x, θ - 1 and φ - 1 can be omitted

$\begin{matrix} [\begin{matrix} \frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1})} + \frac{a_{2}}{x^{'} (a_{2} + b_{2}) + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0 \end{matrix}$

In particular, setting b₁ = b₂ = 0 and expand the inequality leads to the result (x′) ² - (x′ - zφ) zφ ≥ 0 which will be true only if x′ ≥ zφ and no other combinations of a_i′s and b_i′s lead to other conditions on x, x′, y and z . Thus x′ ≥ max(yθ, zφ) as the necessary conditions for the family to be T_L-transitivity.

Corollary 2.1.2. If θ = φ, then T_L-transitive members of the class of inclusion measure (6) are characterized by the necessary conditions $x^{'} \geq max (y θ, z θ)$

Corollary 2.1.3. If θ = 1 = φ, then T_L -transitive members of the class of inclusion measure (6) are characterized by the necessary conditions $x^{'} \geq max (y, z)$

Theorem 2.1.4. The T_L-transitive members of the class of inclusion measures (6) are characterized by the necessary conditions $\begin{matrix} min (x^{'}, t^{'}) & \geq & max (y θ, z φ) \\ \land (x^{'})^{2} & \geq & max (y θ (t^{'} - y θ), z φ (t^{'} - z φ)) \\ \land (t^{'})^{2} & \geq & max (y θ (x^{'} - y θ), z φ (x^{'} - z φ)) \end{matrix}$

Proof. To determine the conditions on parameters x, x′, t, t′, y, and z for transitivity, the inequality (1) can be written in terms of family (6), as

$\begin{matrix} [1 - \frac{x | A ∖ B | + t | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} | A ∖ B | + t^{'} | B ∖ A | + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ + [1 - \frac{x | B ∖ C | + t | C ∖ B | + y | B \cap C | + z | {(B \cup C)}^{c} |}{x^{'} | B ∖ C | + t^{'} | C ∖ B | + y θ | B \cap C | + z φ | {(B \cup C)}^{c} |}] \\ - [1 - \frac{x | A ∖ C | + t | C ∖ A | + y | A \cap C | + z | {(A \cup C)}^{c} |}{x^{'} | A ∖ C | + t^{'} | C ∖ A | + y θ | A \cap C | + z φ | {(A \cup C)}^{c} |}] \geq 0 \end{matrix}$

Considering the settings in figure and using the values of cardinalities given in (7), the above inequality leads to

Setting a₁ = b₁ = c = d = 0 in (8) and ignoring the factors x′ - x, t′ - t, θ - 1 and φ - 1 leads to $[\begin{matrix} \frac{b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (11) and $[\begin{matrix} \frac{a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{3} + b_{2}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (12)

In particular, setting a₂ = a₃ = 0 in inequality (11) implies $(x^{'})^{2} - y θ (t^{'} - y θ) \geq 0$ (13) and setting b₂ = b₃ = 0 in inequality (12) implies $(t^{'})^{2} - z φ (x^{'} - z φ) \geq 0$ (14)

Other combinations of a_i′s and b_i′s do not provide any other condition on x, x′, t, t′, y, z. Setting a₂ = b₂ = c = d = 0 in (10). As x′ > x, t′ > t, θ ≥ 1 and φ ≥ 1, so the factors x′ - x, t′ - t, θ - 1 and φ - 1 can be omitted and this leads to the inequality (15) and the inequality (16), $[\begin{matrix} \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} \end{matrix}] \geq 0$ (15) and $[\begin{matrix} \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{3} + b_{1}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} \end{matrix}] \geq 0$ (16)

In particular, setting a₁ = a₃ = 0 and b₁ = b₃ = 0 in inequality (15) implies t′ - yθ ≥ 0 and t′ - zφ ≥ 0 repectively and a₃ = b₁ = 0 in (15) leads to the weak conditions (x′) ² ≥ yzθφ . Similarly by setting a₁ = a₃ = 0 and b₁ = b₃ = 0 in (16) impliesand a₁ = b₃ = 0 in inequality (16) gives a weaker condition (t′) ² ≥ yzθφ . Thus inequalities (15) and (16) will be true only if t′ ≥ max(yθ, zφ) and x′ ≥ max(yθ, zφ). That is $min (x^{'}, t^{'}) \geq max (y θ, z φ)$ (17)

Setting a₃ = b₃ = c = d = 0 in (10). Using x′ > x, t′ > t, θ ≥ 1 and φ ≥ 1 so the factors x′ - x, t′ - t, θ - 1 and φ - 1 can be omitted and this leads to the inequality (18) and the inequality (19), $[\begin{matrix} \frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{a_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (18) and $[\begin{matrix} \frac{a_{2} + b_{1}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{b_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (19)

In particular, setting a₁ = a₂ = 0 in the inequality (19) implies $(t^{'})^{2} - y θ (x^{'} - y θ) \geq 0$

In particular, setting b₁ = b₂ = 0 in the inequality (18) implies $(x^{'})^{2} - z φ (t^{'} - z φ) \geq 0$ (20)

Other combinations of a_i′s and b_i′s do not provide any other condition on x, x′, t, t′, y, z. Above conditions (13) and (20) can be combine to $(x^{'})^{2} \geq max (y θ (t^{'} - y θ), z φ (t^{'} - z φ))$ and conditions (14) and (2.1) can be combine to $(t^{'})^{2} \geq max (y θ (x^{'} - y θ), z φ (x^{'} - z φ))$

2.2 Product transitive members

Theorem 2.2.1. The T_P-transitive members of the class of inclusion measures (6) are characterized by the necessary conditions

$\begin{matrix} x^{'} \geq max (y θ, z φ) \\ \land x^{'} \geq max (x^{'} θ - z φ, x^{'} φ - y θ) \\ \land {xx}^{'} - {yx}^{'} θ + y^{2} θ \geq 0 \\ \land {xx}^{'} - {zx}^{'} φ + z^{2} φ \geq 0 \\ \land {xx}^{'} + {zx}^{'} - yz φ θ - {zx}^{'} φ \geq 0 \\ \land {xx}^{'} + {yx}^{'} - yz θ φ - {yx}^{'} φ \geq 0 \end{matrix}$

Proof. To determine the conditions on parameters x, x′, y, and z for transitivity, the inclusion measure I is T_P-transitive if it holds the inequality (2). Then the inequality (2) can be written in terms of inclusion measure (6),

$\begin{matrix} [1 - \frac{x | A ∖ B | + x^{'} | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} (| A ∖ B | + | B ∖ A |) + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ + [1 - \frac{x | B ∖ C | + x^{'} | C ∖ B | + y | B \cap C | + z | {(B \cup C)}^{c} |}{x^{'} (| B ∖ C | + | C ∖ B |) + y θ | B \cap C | + z φ | {(B \cup C)}^{c} |}] \\ - [1 - \frac{x | A ∖ B | + x^{'} | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} (| A ∖ B | + | B ∖ A |) + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ \cdot [1 - \frac{x | B ∖ C | + x^{'} | C ∖ B | + y | B \cap C | + z | {(B \cup C)}^{c} |}{x^{'} (| B ∖ C | + | C ∖ B |) + y θ | B \cap C | + z φ | {(B \cup C)}^{c} |}] \\ - [1 - \frac{x | A ∖ C | + x^{'} | C ∖ A | + y | A \cap C | + z | {(A \cup C)}^{c} |}{x^{'} (| A ∖ C | + | C ∖ A |) + y θ | A \cap C | + z φ | {(A \cup C)}^{c} |}] \geq 0 \end{matrix}$

Using the values of cardinalities given in (7), the above inequality implies

$\begin{matrix} [1 - \frac{x (a_{1} + b_{2}) + x^{'} (a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ + [1 - \frac{x (a_{2} + b_{3}) + x^{'} (a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{2}) + x^{'} (a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ \cdot [1 - \frac{x (a_{2} + b_{3}) + x^{'} (a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{3}) + x^{'} (a_{3} + b_{1}) + y (b_{2} + c) + z (a_{2} + d)}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1}) + y θ (b_{2} + c) + z φ (a_{2} + d)}] \geq 0 \end{matrix}$ (21)

Setting a₁ = b₁ = c = d = 0, (21) implies

$\begin{matrix} (x^{'} - x) [\begin{matrix} \frac{b_{2}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2})} \\ - \frac{b_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} - (\frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2})}) \\ \cdot (\frac{(x^{'} - x) b_{2} + y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} (a_{2} + b_{2}) + y θ b_{3} + z φ a_{3}}) \end{matrix}] \\ + y (θ - 1) [\begin{matrix} \frac{b_{3}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} \\ - \frac{b_{2}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} \end{matrix}] \\ + z (φ - 1) [\begin{matrix} \frac{a_{3}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} \\ - \frac{a_{2}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0 \end{matrix}$

Since x′ > x, θ ≥ 1 and φ ≥ 1, so the factors x′ - x, θ - 1 and φ - 1 can be omitted and we obtain

$\begin{matrix} [\begin{matrix} \frac{b_{2}}{x^{'} (b_{2} + a_{2}) + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2})} \\ - \frac{b_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{2} + z φ a_{2}} - (\frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2})}) \\ \cdot (\frac{(x^{'} - x) b_{2} + y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} (a_{2} + b_{2}) + y θ b_{3} + z φ a_{3}}) \end{matrix}] \geq 0 \end{matrix}$

Expanding the above inequality and setting a₂ = a₃ = 0 leads to

${xx}^{'} + y^{2} θ - {yx}^{'} θ \geq 0$ (22)

and setting a₃ = b₂ = 0 leads to

$x^{'} + z φ - x^{'} θ \geq 0$ (23)

Other combinations of a_i′s and b_i′s, that is, b₂ = b₃ = 0 or a₂ = b₃ = 0 do not lead to any other conditions on x, x′, y, z. Setting a₂ = b₂ = c = d = 0, in (21) and ignoring the factors x′ - x, θ - 1 and φ - 1, we obtain

$\begin{matrix} [\begin{matrix} \frac{a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1})} \\ + \frac{b_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{1} + z φ a_{1}} - (\frac{a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}}) \\ \cdot (\frac{(x^{'} - x) b_{3} + y (θ - 1) b_{1} + z (φ - 1) a_{1}}{x^{'} (a_{3} + b_{3}) + y θ b_{1} + z φ a_{1}}) \\ - (\frac{b_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{1} + z φ a_{1}}) \cdot (\frac{y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}}) \end{matrix}] \\ \geq 0 \end{matrix}$ (24)

In particular, setting a₁ = a₃ = 0 in inequality (24) leads to the result

$(x^{'} - y θ) (y θ b_{1} b_{3}^{2} + x^{'} b_{1}^{2} b_{3}) \geq 0$ (25) which is true if x′ ≥ yθ. Similarly, setting of b₁ = b₃ = 0 in inequality (24) implies $(x^{'} - z φ) (x^{'} a_{1} a_{3}^{2} + z φ a_{1}^{2} a_{3}) \geq 0$ (26)

that is, x′ ≥ zφ is necessary to hold the inequality and setting of b₁ = a₃ = 0 in (24) ends up with a condition

$\begin{matrix} ({zx}^{'} + {xx}^{'} - yz θ φ - {zx}^{'} φ) a_{1}^{2} b_{3} \\ + ({yx}^{'} + {xx}^{'} - yz θ φ - {yx}^{'} φ) a_{1} b_{3}^{2} \geq 0 . \end{matrix}$ that is, ${zx}^{'} + {xx}^{'} - yz θ φ - {zx}^{'} φ \geq 0$ (27) $\land {yx}^{'} + {xx}^{'} - yz θ φ - {yx}^{'} φ \geq 0 .$ (28)

Other combinations of $a_{i}^{'} s$ and $b_{i}^{'} s$ do not lead to any other conditions on x, x′, y, z. Setting a₃ = b₃ = c = d = 0 in (21) and factors x′ - x, θ - 1 and φ - 1 can be omitted. Thus $\begin{matrix} [\begin{matrix} \frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1})} + \frac{a_{2}}{x^{'} (a_{2} + b_{2}) + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{2} + z φ a_{2}} - (\frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1})}) \\ \cdot (\frac{(x^{'} - x) a_{2} + y (θ - 1) b_{1} + z (φ - 1) a_{1}}{x^{'} (a_{2} + b_{2}) + y θ b_{1} + z φ a_{1}}) \end{matrix}] \geq 0 \end{matrix}$

In particular, setting b₁ = b₂ = 0 and expand the inequality leads to the result ${xx}^{'} + z^{2} φ - x^{'} z φ \geq 0$ (29) and setting a₂ = b₁ = 0 leads to the result $x^{'} + y θ - x^{'} φ \geq 0$ (30)

Other combinations of a_i′s and b_i′s do not lead to other conditions on x, x′, y and z.

Theorem 2.2.2. The T_P-transitive members of the class of inclusion measures (5) are characterized by the necessary conditions

$\begin{matrix} x^{'} - y θ \geq 0 \\ \land {xx}^{'} + y^{2} θ - {yt}^{'} θ \geq 0 \\ \land {xx}^{'} + z^{2} φ - {zt}^{'} φ \geq 0 \\ \land {tt}^{'} + y^{2} θ - {yx}^{'} θ \geq 0 \\ \land {tt}^{'} + z^{2} φ - {zx}^{'} φ \geq 0 \\ \land {xx}^{'} + {yx}^{'} - x^{'} y θ - yz θ φ \geq 0 \\ \land {xx}^{'} + {zx}^{'} - x^{'} z φ - yz θ φ \geq 0 \\ \land {tt}^{'} + {yt}^{'} - t^{'} y θ - yz θ φ \geq 0 \\ \land {tt}^{'} + {zt}^{'} - t^{'} z φ - yz θ φ \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + {yt}^{'} - y^{2} θ^{2} \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + {zt}^{'} - z^{2} φ^{2} \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + yz φ + tz φ \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + yz θ + ty θ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + {zx}^{'} - z^{2} φ^{2} \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + yz θ + xy θ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + xz φ + yz φ \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + {yx}^{'} - {yx}^{'} θ + yz φ \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + {zx}^{'} - {zx}^{'} φ + yz θ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + {zt}^{'} - {zt}^{'} φ + yz θ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + {yt}^{'} - {yt}^{'} θ + yz φ \geq 0 \end{matrix}$

Proof. To determine the conditions on parameters x, x′, y, and z for transitivity, let us suppose that the inclusion measure S is T_P-transitive i.e., $\begin{matrix} [1 - \frac{x | A ∖ B | + t | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} | A ∖ B | + t^{'} | B ∖ A | + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ + [1 - \frac{x | B ∖ C | + t | C ∖ B | + y | B \cap C | + z | {(B \cup C)}^{c} |}{x^{'} | B ∖ C | + t^{'} | C ∖ B | + y θ | B \cap C | + z φ | {(B \cup C)}^{c} |}] \\ - [1 - \frac{x | A ∖ B | + t | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} | A ∖ B | + t^{'} | B ∖ A | + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ \cdot [1 - \frac{x | B ∖ C | + t | C ∖ B | + y | B \cap C | + z | {(B \cup C)}^{c} |}{x^{'} | B ∖ C | + t^{'} | C ∖ B | + y θ | B \cap C | + z φ | {(B \cup C)}^{c} |}] \\ - [1 - \frac{x | A ∖ C | + t | C ∖ A | + y | A \cap C | + z | {(A \cup C)}^{c} |}{x^{'} | A ∖ C | + t^{'} | C ∖ A | + y θ | A \cap C | + z φ | {(A \cup C)}^{c} |}] \geq 0 \end{matrix}$

Considering the settings in figure and using the values of cardinalities given in (7), the above inequality leads to

$\begin{matrix} [1 - \frac{x (a_{1} + b_{2}) + t (a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ + [1 - \frac{x (a_{2} + b_{3}) + t (a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{2}) + t (a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ \cdot [1 - \frac{x (a_{2} + b_{3}) + t (a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{3}) + t (a_{3} + b_{1}) + y (b_{2} + c) + z (a_{2} + d)}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1}) + y θ (b_{2} + c) + z φ (a_{2} + d)}] \geq 0 \end{matrix}$ (31)

Setting a₁ = b₁ = c = d = 0 in (31) and ignoring the factors x′ - x, t′ - t, θ - 1 and φ - 1, the inequality can be simplified as (32) $[\begin{matrix} \frac{b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} - (\frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ \cdot (\frac{(x^{'} - x) b_{2} + (t^{'} - t) a_{2} + y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \end{matrix}] \geq 0$ (32) and as (33) $[\begin{matrix} \frac{a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{3} + b_{2}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} - (\frac{a_{3} + b_{2}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ \cdot (\frac{(x^{'} - x) b_{2} + (t^{'} - t) a_{2} + y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \end{matrix}] \geq 0$ (33)

In particular, setting a₂ = a₃ = 0 in inequality (34) implies

${xx}^{'} + y^{2} θ - {yt}^{'} θ \geq 0$ (34) and setting b₂ = a₃ = 0 in inequality (35) implies

${yx}^{'} + {tx}^{'} - x^{'} t^{'} - {yx}^{'} θ + yz φ \geq 0$ (35) $\land {tx}^{'} - x^{'} t^{'} + yz φ + tz φ \geq 0$ (36) and setting b₂ = b₃ = 0 in inequality (37) implies ${tt}^{'} - {zx}^{'} φ + z^{2} φ \geq 0$ (37) and setting a₂ = b₃ = 0 in inequality (33) implies ${xt}^{'} + {zt}^{'} - x^{'} t^{'} - {zt}^{'} φ + yz θ \geq 0$ (38) $\land {xt}^{'} - x^{'} t^{'} + yz θ + xy θ \geq 0$ (39)

Other combinations of a_i′s and b_i′s do not provide any other condition on x, x′, t, t′, y, z. Setting a₂ = b₂ = c = d = 0 (31) and the factors x′ - x, t′ - t, θ - 1 and φ - 1 can be omitted and this leads to the inequality (40) and the inequality (41),

$[\begin{matrix} \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} - (\frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}}) \\ \cdot (\frac{(x^{'} - x) b_{3} + (t^{'} - t) a_{3} + y (θ - 1) b_{1} + z (φ - 1) a_{1}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}}) \\ - (\frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}}) \cdot (\frac{(t^{'} - t) b_{1} + y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}}) \end{matrix}] \geq 0$ (40) and $[\begin{matrix} \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{3} + b_{1}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} - (\frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}}) \\ \cdot (\frac{(t^{'} - t) a_{3} + y (θ - 1) b_{1} + z (φ - 1) a_{1}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}}) \\ - (\frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}}) \cdot (\frac{(x^{'} - x) a_{1} + y (θ - 1) b_{3} + z (φ - 1) a_{3}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}}) \end{matrix}] \geq 0$ (41)

In particular, setting a₁ = a₃ = 0, b₁ = b₃ = 0 and a₃ = b₁ = 0 in inequality (40) implies $\begin{matrix} {tx}^{'} + {yt}^{'} - x^{'} t^{'} - y^{2} θ^{2} \geq 0 \\ \land {tx}^{'} + {zt}^{'} - x^{'} t^{'} - z^{2} φ^{2} \geq 0 \\ \land {yx}^{'} + {xx}^{'} - x^{'} y θ - yz θ φ \geq 0 \\ \land {zx}^{'} + {xx}^{'} - x^{'} z φ - yz θ φ \geq 0 \end{matrix}$

Similarly by setting a₁ = a₃ = 0, b₁ = b₃ = 0 and a₁ = b₃ = 0 in (41) implies $\begin{matrix} x^{'} - y θ \geq 0 \\ \land {xt}^{'} + {zx}^{'} - x^{'} t^{'} - z^{2} φ^{2} \geq 0 \\ \land {yt}^{'} + {tt}^{'} - t^{'} y θ - yz θ φ \geq 0 \\ \land {zt}^{'} + {tt}^{'} - t^{'} z φ - yz θ φ \geq 0 \end{matrix}$

Setting a₃ = b₃ = c = d = 0 (31) and the factors x′ - x, t′ - t, θ - 1 and φ - 1 can be omitted and this leads to the inequality (42) and the inequality (43), $[\begin{matrix} \frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{a_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} - (\frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})}) \\ \cdot (\frac{(x^{'} - x) a_{2} + (t^{'} - t) b_{2} + y (θ - 1) b_{1} + z (φ - 1) a_{1}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}}) \end{matrix}] \geq 0$ (42)

and $[\begin{matrix} \frac{a_{2} + b_{1}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{b_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} - (\frac{a_{2} + b_{1}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})}) \\ \cdot (\frac{(x^{'} - x) a_{2} + (t^{'} - t) b_{2} + y (θ - 1) b_{1} + z (φ - 1) a_{1}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}}) \end{matrix}] \geq 0$ (43)

In particular, setting a₁ = a₂ = 0 in the inequality (43) implies ${tt}^{'} - {yx}^{'} θ + y^{2} θ \geq 0$ (44) and setting a₁ = b₂ = 0 in the inequality (43) implies ${xt}^{'} - x^{'} t^{'} + xz φ + yz φ \geq 0$ (45) ${xt}^{'} + {yt}^{'} - x^{'} t^{'} - {yt}^{'} θ + yz φ \geq 0$ (46)

In particular, setting b₁ = b₂ = 0 in the inequality (42) implies ${xx}^{'} - {zt}^{'} φ + z^{2} φ \geq 0$ (47) and setting a₂ = b₁ = 0 in the inequality (42) implies ${tx}^{'} - x^{'} t^{'} + yz θ + ty θ \geq 0$ (48) ${tx}^{'} + {zx}^{'} - x^{'} t^{'} - {zx}^{'} φ + yz θ \geq 0$ (49)

Other combinations of a_i′s and b_i′s do not provide any other condition on x, x′, t, t′, y, z.

2.3 Min transitive members

Theorem 2.3.1. The family (6) does not contain any T_M-transitive members.

Proof. The inclusion measure I is T_M-transitive if it fulfill the definition of transitivity (3). For this purpose, we have to verify that either I (A, B) ≤ I (A, C) or I (B, C) ≤ I (A, C). That is,

$\begin{matrix} [1 - \frac{x | A ∖ B | + x^{'} | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} (| A ∖ B | + | B ∖ A |) + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ - [1 - \frac{x | A ∖ C | + x^{'} | C ∖ A | + y | A \cap C | + z | {(A \cup C)}^{c} |}{x^{'} (| A ∖ C | + | C ∖ A |) + y θ | A \cap C | + z φ | {(A \cup C)}^{c} |}] \geq 0 \end{matrix}$

Using the values of cardinalities given in (7), the above inequality implies

$\begin{matrix} [1 - \frac{x (a_{1} + b_{2}) + x^{'} (a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ - [1 - \frac{x (a_{1} + b_{3}) + x^{'} (a_{3} + b_{1}) + y (b_{2} + c) + z (a_{2} + d)}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1}) + y θ (b_{2} + c) + z φ (a_{2} + d)}] \geq 0 \end{matrix}$ (50)

Setting a₂ = b₂ = c = d = 0 in (42) and ignoring the factors x′ - x, θ - 1 and φ - 1, the inequality can be simplified as

$\begin{matrix} \frac{a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} \\ - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1}) \geq 0} \end{matrix}$ (51)

On simplification, this leads to contradiction as

$\begin{matrix} - \frac{x^{'} b_{1} b_{3} - x^{'} a_{1} a_{3} + y θ b_{2}^{2} + y θ a_{1} b_{2} + z φ a_{1} a_{2} + z φ a_{2} b_{2}}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1}) (x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3})} \\ \geq 0 \end{matrix}$ (52)

Remark 2.3.2. The family (5) does not contain any T_M transitive members.

3 Transitivity of similarity measures

A common approach for comparing two objects is to select an appropriate list of features then prepare a binary vector of {0, 1} based on the fact that if the object has certain feature will be recorded as 1 and 0 otherwise. The degree of similarity of the two objects is then expressed in terms of cardinalities of these binary vector sets A and B in a finite universe X as $S (A, B) = \frac{x α_{A, B} + t ω_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} α_{A, B} + t^{'} ω_{A, B} + y^{'} δ_{A, B} + z^{'} ν_{A, B}}$ (53) with A, B ∈ P (X) , ω_A,B = max(|A ∖ B|, |B ∖ A|) , α_A,B = min(|A ∖ B|, |B ∖ A|) , δ_A,B = |A ∩ B|, ν_A,B = | (A ∪ B) ^c| and parameters x, x′, t, t′, y, y′, z, z′ ∈ {0, 1} . The similarity measure is symmetric, that is, for all A, B ∈ F (X) , S (A, B) = S (B, A) and fuzzy reflexive for any A ∈ F (X) , S (A, A) ∈ [0, 1]. Janssens et al. [22] used four parameters to test the various properties of monotonicity and transitivity for Łukasiewicz, Product and Min t-norms. Some members of the family (53) with parameters discussed by Janssens [22] in the light of eight parameters are given in Table 2. Many other measures cannot be discussed under parametric family of similarity measures using four parameters, some of them are given in Table 3. Some of these measures are explored in this section by using all eight parameters. Fuzzy similarity measure will hold fuzzy reflexivity for y′ = yθ, z′ = zφ with θ, φ ≥ 1, and the above relation of similarity measure can be written as

$S (A, B) = \frac{x α_{A, B} + t ω_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} α_{A, B} + t^{'} ω_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}$ (54)

For simplicity, if we use only four parameters, that is, x = t, x′ = t′, the above relation becomes $S (A, B) = \frac{x Δ_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} Δ_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}$ (55) which is an extension of four parameters of Janssens’ to six parameters with _ΔA,B = |AΔB|. $S (A, B) = \frac{x Δ_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} Δ_{A, B} + y δ_{A, B} + z ν_{A, B}}$ (56)

These similarity measures are symmetric for any fuzzy subsets A, B and will be fuzzy reflexive if y′ = yθ and z′ = zφ for θ, φ ≥ 1 . That is with positive real parameters x, x′, t, t′, y, z . S (A, B) ∈ [0, 1] if 0 ≤ x < x′ and 0 ≤ t < t′. This section is focused to characterize the Łukasiewicz-transitive members of family (53).

3.1 Lukasiewicz transitive members

Theorem 3.1.1. The T_L-transitive members of family (equ3.03) are characterized by the necessary condition $x^{'} \geq max (y θ, z φ) .$

Proof. To determine the conditions on parameters x, x′, y, and z for transitivity, the similarity measure S is T_L-transitive if it holds the inequality (equ1.1). In terms of family (equ3.03), the inequality can be written as

\begin{matrix} [1 - \frac{x Δ_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} Δ_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}] \\ + [1 - \frac{x Δ_{B, C} + y δ_{B, C} + z ν_{B, C}}{x^{'} Δ_{B, C} + y θ δ_{B, C} + z φ ν_{B, C}}] \\ - [\frac{x Δ_{A, C} + y δ_{A, C} + z ν_{A, C}}{x^{'} Δ_{A, C} + y θ δ_{A, C} + z φ ν_{A, C}}] \geq 0 \end{matrix}

Consider the setting in figure, and substituting the cardinalities (equ2.04), the above inequality implies

Similarly setting a₂ = b₂ = c = d = 0 and omitting the factors x′ - x, θ - 1 and φ - 1, the inequality (equ31.11) implies that

$\begin{matrix} [\begin{matrix} \frac{b_{1} + a_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} \\ + \frac{b_{3} + a_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{1} + z φ a_{1}} - \frac{1}{x^{'}} \end{matrix}] \geq 0 \\ [\begin{matrix} ((x^{'})^{2} - z^{2} φ^{2}) a_{1} a_{3} + ((x^{'})^{2} - y^{2} θ^{2}) b_{1} b_{3} \\ + ((x^{'})^{2} - yz θ φ) (a_{1} b_{3} + a_{3} b_{1}) \end{matrix}] \\ \geq 0 \end{matrix}$ (58)

Now discussing the above inequality, by setting a₁ = a₃ = 0, the inequality leads to the result (x′) ² - y²θ² ≥ 0 which is fulfilled only if x′ ≥ yθ and setting b₁ = b₃ = 0, the inequality leads to (x′) ² - z²φ² ≥ 0, which is satisfied if x′ ≥ zφ. Similarly, setting a₁ = b₃ = 0 or b₁ = a₃ = 0 in the inequality leads to the weaker condition (x′) ² - yzθφ ≥ 0. These conditions can be combine into a single condition as $x^{'} \geq max (y θ, z φ) .$ (59)

Setting a₁ = b₁ = c = d = 0, and a₃ = b₃ = c = d = 0 the inequality (equ31.11) does not imply any further condition on x, x′, y and z.

Theorem 3.1.2. The T_L-transitive members of family (equ3.02) are characterized by the necessary condition $\begin{matrix} min (x^{'}, t^{'}) & \geq & max (y θ, z φ) \\ \land (t^{'})^{2} & \geq & max ((x^{'} - y θ) y θ, (x^{'} - z φ) z φ) \\ \land (x^{'})^{2} & \geq & max ((t^{'} - y θ) y θ, (t^{'} - z φ) z φ) \end{matrix}$

Proof. To determine the conditions on parameters x, x′, t, t′, y, and z for transitivity, the similarity measure S is T_L-transitive if it holds In terms of family (equ3.02), the above inequality can be written as $\begin{matrix} [1 - \frac{x | A ∖ B | + t | B ∖ A | + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} | A ∖ B | + t^{'} | B ∖ A | + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ + [1 - \frac{x | B ∖ C | + t | C ∖ B | + y | B \cap C | + z | {(B \cup C)}^{c} |}{x^{'} | B ∖ C | + t^{'} | C ∖ B | + y θ | B \cap C | + z φ | {(B \cup C)}^{c} |}] \\ - [1 - \frac{x | A ∖ C | + t | C ∖ A | + y | A \cap C | + z | {(A \cup C)}^{c} |}{x^{'} | A ∖ C | + t^{'} | C ∖ A | + y θ | A \cap C | + z φ | {(A \cup C)}^{c} |}] \geq 0 \end{matrix}$

Consider the setting in figure, and substituting the cardinalities (equ2.04), the above inequality implies $\begin{matrix} [1 - \frac{x (a_{1} + b_{2}) + t (a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ + [1 - \frac{x (a_{2} + b_{3}) + t (a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{3}) + t (a_{3} + b_{1}) + y (b_{2} + c) + z (a_{2} + d)}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1}) + y θ (b_{2} + c) + z φ (a_{2} + d)}] \geq 0 \end{matrix}$ (60)

Setting a₁ = b₁ = c = d = 0 in the inequality (equ31.21) and ignoring the factors x′ - x, θ - 1 and φ - 1, we obtain $[\begin{matrix} \frac{b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (61) and $[\begin{matrix} \frac{a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{3} + b_{2}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (62)

In particular, setting a₂ = a₃ = 0, the inequality (equ31.22) leads to the results $(x^{'})^{2} - (t^{'} - y θ) y θ \geq 0$ (63) and setting b₂ = b₃ = 0, the inequality (equ31.23) leads to the results $(t^{'})^{2} - (x^{'} - z φ) z φ \geq 0$ (64)

Other combinations of a_i′s and b_i′s do not provide any other condition on x, x′, t, t′, y, z. Similarly setting a₂ = b₂ = c = d = 0, the inequality (equ31.21) implies that $[\begin{matrix} \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} \end{matrix}] \geq 0$ (65) and $[\begin{matrix} \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{3} + b_{1}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} \end{matrix}] \geq 0$ (66)

In particular, setting a₁ = a₃ = 0 and b₁ = b₃ = 0 the inequality (equ31.26) leads to the results t′ - yθ ≥ 0 and t′ - zφ ≥ 0 respectively, which can be written in the form $t^{'} \geq max (y θ, z φ)$ (67)

Also setting a₁ = a₃ = 0 and b₁ = b₃ = 0, the inequality (equ31.27) leads to the results x′ - yθ ≥ 0 and x′ - zφ ≥ 0 respectively, which can be combine in the form $x^{'} \geq max (y θ, z φ)$ (68)

The results (equ31.28) and (equ31.29) leads to $min (x^{'}, t^{'}) \geq max (y θ, z φ)$ (69)

Similarly setting a₃ = b₃ = c = d = 0, the inequality (equ31.21) implies that $[\begin{matrix} \frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{a_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (70) and $[\begin{matrix} \frac{a_{2} + b_{1}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{b_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} \end{matrix}] \geq 0$ (71)

In particular, setting a₁ = a₂ = 0,the inequality (equ31.212) leads to the result ${(t^{'})}^{2} - (x^{'} - y θ) y θ \geq 0$ (72)

and setting b₁ = b₂ = 0,the inequality (equ31.211) leads to the result $(x^{'})^{2} - (t^{'} - z φ) z φ \geq 0$ (73)

Now results (equ31.24) and (equ31.214) leads to $(x^{'})^{2} \geq max ((t^{'} - y θ) y θ, (t^{'} - z φ) z φ)$ (74)

and results (equ31.25) and (equ31.213) merged to $(t^{'})^{2} \geq max ((x^{'} - y θ) y θ, (x^{'} - z φ) z φ)$ (75)

Thus the necessary condition for transitivity is obtained by (equ31.210, equ31.215 and equ31.216) as $\begin{matrix} min (x^{'}, t^{'}) & \geq & max (y θ, z φ) \\ \land (t^{'})^{2} & \geq & max ((x^{'} - y θ) y θ, (x^{'} - z φ) z φ) \\ \land (x^{'})^{2} & \geq & max ((t^{'} - y θ) y θ, (t^{'} - z φ) z φ) \end{matrix}$

3.2 Product transitive members

Theorem 3.2.1. The T_P-transitive members of family (equ3.03) are characterized by the necessary condition ${xx}^{'} \geq max (yz θ φ, y^{2} θ^{2}, z^{2} φ^{2})$

Proof. To determine the conditions on parameters x, x′, y,and z for transitivity, the similarity measure S is T_P-transitive if it holds the inequality (equ1.2). In terms of family (equ3.03), the inequality (equ1.2) can be written as $\begin{matrix} [1 - \frac{x Δ_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} Δ_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}] \\ + [1 - \frac{x Δ_{B, C} + y δ_{B, C} + z ν_{B, C}}{x^{'} Δ_{B, C} + y θ δ_{B, C} + z φ ν_{B, C}}] \\ - [1 - \frac{x Δ_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} Δ_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}] \\ \cdot [1 - \frac{x Δ_{B, C} + y δ_{B, C} + z ν_{B, C}}{x^{'} Δ_{B, C} + y θ δ_{B, C} + z φ ν_{B, C}}] \\ - [1 - \frac{x Δ_{A, C} + y δ_{A, C} + z ν_{A, C}}{x^{'} Δ_{A, C} + y θ δ_{A, C} + z φ ν_{A, C}}] \geq 0 \end{matrix}$

Consider the setting in figure, and substituting the cardinalities (equ2.04), the above inequality implies

$\begin{matrix} [1 - \frac{x (a_{1} + b_{2} + a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ + [1 - \frac{x (a_{2} + b_{3} + a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{2} + a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ \cdot [1 - \frac{x (a_{2} + b_{3} + a_{3} + b_{2}) + y (b_{1} + c) + z (a_{1} + d)}{x^{'} (a_{2} + b_{3} + a_{3} + b_{2}) + y θ (b_{1} + c) + z φ (a_{1} + d)}] \\ - [1 - \frac{x (a_{1} + b_{3} + a_{3} + b_{1}) + y (b_{2} + c) + z (a_{2} + d)}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1}) + y θ (b_{2} + c) + z φ (a_{2} + d)}] \geq 0 \end{matrix}$ (76)

Setting a₂ = b₂ = c = d = 0, the inequality (equ32.11) implies that

$\begin{matrix} [\begin{matrix} \frac{(a_{1} + b_{1})}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} + \frac{b_{3} + a_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{1} + z φ a_{1}} - \frac{1}{x^{'}} \\ - (\frac{(x^{'} - x) (a_{1} + b_{1})}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}}) \cdot (\frac{b_{3} + a_{3}}{x^{'} (b_{3} + a_{3}) + y θ b_{1} + z φ a_{1}}) \end{matrix}] \\ \geq 0 \end{matrix}$ (77)

In particular, setting a₁ = a₃ = 0,the above inequality leads to the result xx′ - y²θ² ≥ 0 which is fulfilled only if xx′ ≥ y²θ² and setting b₁ = b₃ = 0,the inequality leads to the result xx′ - z²φ² ≥ 0 which is satisfied if xx′ ≥ z²φ². Similarly, setting a₁ = b₃ = 0,and b₁ = a₃ = 0,in the inequality as xx′ - yzθφ ≥ 0 lead to the result xx′ ≥ yzθφ. These conditions can be combine into a single condition as ${xx}^{'} \geq max (yz θ φ, y^{2} θ^{2}, z^{2} φ^{2}) .$ (78)

Setting a₁ = b₁ = c = d = 0,and a₃ = b₃ = c = d = 0 the inequality (equ32.11) does not imply any further condition on x, x′, y and z.

Theorem 3.2.2. The T_P-transitive members of family (equ3.02) are characterized by the necessary condition $\begin{matrix} min (x^{'}, t^{'}) \geq max (y θ, z φ) \\ \land min ({xx}^{'}, {tt}^{'}) \geq yz θ φ \\ \land {tx}^{'} - x^{'} t^{'} + yz θ φ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + yz θ φ \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + tz φ + yz θ φ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + xy θ + yz θ φ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + {yx}^{'} θ - y^{2} θ^{2} \geq 0 \\ \land {tx}^{'} - x^{'} t^{'} + xy θ + {yx}^{'} θ \geq 0 \\ \land {xt}^{'} - x^{'} t^{'} + tz φ + {zt}^{'} φ \geq 0 \end{matrix}$

Proof. To determine the conditions on parameters x, x′, t, t′, y,and z for transitivity, the similarity measure S is T_P-transitive if it fulfill the condition (equ1.2) which can be written in terms of family (equ3.02), as

$\begin{matrix} [1 - \frac{x α_{A, B} + t ω_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} α_{A, B} + t^{'} ω_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}] \\ + [1 - \frac{x α_{B, C} + t ω_{B, C} + y δ_{B, C} + z ν_{B, C}}{x^{'} α_{B, C} + t^{'} ω_{B, C} + y θ δ_{B, C} + z φ ν_{B, C}}] \\ - [1 - \frac{x α_{A, B} + t ω_{A, B} + y δ_{A, B} + z ν_{A, B}}{x^{'} α_{A, B} + t^{'} ω_{A, B} + y θ δ_{A, B} + z φ ν_{A, B}}] \\ \cdot [1 - \frac{x α_{B, C} + t ω_{B, C} + y δ_{B, C} + z ν_{B, C}}{x^{'} α_{B, C} + t^{'} ω_{B, C} + y θ δ_{B, C} + z φ ν_{B, C}}] \\ - [1 - \frac{x α_{A, C} + t ω_{A, C} + y δ_{A, C} + z ν_{A, C}}{x^{'} α_{A, C} + t^{'} ω_{A, C} + y θ δ_{A, C} + z φ ν_{A, C}}] \geq 0 \end{matrix}$

Consider the setting in figure, and substituting the cardinalities (equ2.04), the above inequality implies

Setting a₁ = b₁ = c = d = 0 and omitting the factors x′ - x, t′ - t, θ - 1 and φ - 1 the inequality (equ32.21) leads to

$\begin{matrix} [\begin{matrix} \frac{b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - (\frac{(x^{'} - x) b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{a_{2} + b_{3}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ - (\frac{b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{(t^{'} - t) (a_{3} + b_{2})}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ - (\frac{(t^{'} - t) a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{(a_{2} + b_{3})}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ - \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} \end{matrix}] \\ \geq 0 \end{matrix}$ (80) and

$\begin{matrix} [\begin{matrix} \frac{a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}} + \frac{a_{3} + b_{2}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})} \\ - (\frac{(x^{'} - x) b_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{a_{3} + b_{2}}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ - (\frac{a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{(x^{'} - x) (a_{2} + b_{3})}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ - (\frac{a_{2}}{x^{'} b_{2} + t^{'} a_{2} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{(t^{'} - t) (a_{3} + b_{2})}{x^{'} (a_{2} + b_{3}) + t^{'} (a_{3} + b_{2})}) \\ - \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{2} + z φ a_{2}} \end{matrix}] \\ \geq 0 \end{matrix}$ (81)

In particular, setting a₂ = a₃ = 0,the inequality (equ32.22) leads to the results $\begin{matrix} (xy θ - x^{'} t^{'} + {tx}^{'} + {yx}^{'} θ) b_{2}^{2} b_{3} \\ + (y^{2} θ^{2} + {xx}^{'} - {yt}^{'} θ) b_{2} b_{3}^{2} \geq 0 \\ {xx}^{'} - (t^{'} - y θ) y θ \geq 0 \end{matrix}$ (82) $\land ({tx}^{'} - x^{'} t^{'} + xy θ + {yx}^{'} θ) \geq 0$ (83) and setting b₂ = a₃ = 0,the inequality (equ32.22) leads to $\begin{matrix} (tz φ - x^{'} t^{'} + {tx}^{'} + yz θ φ) a_{2}^{2} b_{3} \\ + ({tx}^{'} - x^{'} t^{'} + yz θ φ) a_{2} b_{3}^{2} \geq 0 \\ {tx}^{'} - x^{'} t^{'} + tz φ + yz θ φ \geq 0 \end{matrix}$ (84) ${tx}^{'} - x^{'} t^{'} + yz θ φ \geq 0$ (85)

and setting b₂ = b₃ = 0,the inequality (equ32.23) leads to the results $\begin{matrix} (z^{2} φ^{2} + {tt}^{'} - {zx}^{'} φ) a_{2} a_{3}^{2} \\ + ({zt}^{'} φ + {xt}^{'} - x^{'} t^{'} + tz φ) a_{2}^{2} a_{3} \geq 0 \\ {tt}^{'} - (x^{'} - z φ) z φ \geq 0 \end{matrix}$ (86) $\land ({xt}^{'} - x^{'} t^{'} + tz φ + {zt}^{'} φ) \geq 0$ (87)

and setting a₂ = b₃ = 0,the inequality (equ32.23) leads to $\begin{matrix} ({xt}^{'} + xy θ + yz θ φ - x^{'} t^{'}) a_{3} b_{2}^{2} \\ + (yz θ φ - x^{'} t^{'} + {xt}^{'}) a_{3}^{2} b_{2} \\ {xt}^{'} + xy θ + yz θ φ - x^{'} t^{'} \geq 0 \end{matrix}$ (88) $yz θ φ - x^{'} t^{'} + {xt}^{'} \geq 0$ (89)

Other combinations of a_i′s and b_i′s do not provide any other condition on x, x′, t, t′, y, z. Similarly setting a₂ = b₂ = c = d = 0 in the inequality (equ32.21), we get

$\begin{matrix} [\begin{matrix} \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{b_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1} + b_{3}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} \\ - (\frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{(x^{'} - x) b_{3} + (t^{'} - t) a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}}) \end{matrix}] \\ \geq 0 \end{matrix}$ (90) and $\begin{matrix} [\begin{matrix} \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}} + \frac{a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{3} + b_{1}}{x^{'} (a_{1} + b_{3}) + t^{'} (a_{3} + b_{1})} \\ - (\frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{3} + z φ a_{3}}) \cdot (\frac{(x^{'} - x) b_{3} + (t^{'} - t) a_{3}}{x^{'} b_{3} + t^{'} a_{3} + y θ b_{1} + z φ a_{1}}) \end{matrix}] \\ \geq 0 \end{matrix}$ (91)

In particular, setting a₁ = a₃ = 0 and b₁ = b₃ = 0 the inequality (equ32.212) leads to the results

$\begin{matrix} [\frac{b_{1}}{t^{'} b_{1} + y θ b_{3}} - \frac{b_{1}}{x^{'} b_{3} + t^{'} b_{1}} - (\frac{b_{1}}{t^{'} b_{1} + y θ b_{3}}) \cdot (\frac{(x^{'} - x) b_{3}}{x^{'} b_{3} + y θ b_{1}})] \geq 0 \\ {xt}^{'} + {yx}^{'} θ - x^{'} t^{'} - y^{2} θ^{2} \geq 0 \end{matrix}$ (92) and $\begin{matrix} [\frac{a_{1}}{x^{'} a_{1} + z φ a_{3}} - \frac{a_{1}}{x^{'} a_{1} + t^{'} a_{3}} - (\frac{a_{1}}{x^{'} a_{1} + z φ a_{3}}) \cdot (\frac{(t^{'} - t) a_{3}}{t^{'} a_{3} + z φ a_{1}})] \geq 0 \\ {tx}^{'} + {zt}^{'} φ - x^{'} t^{'} - z^{2} φ^{2} \geq 0 \end{matrix}$ (93) respectively. Similarly setting a₁ = b₃ = 0 provides $\begin{matrix} [\begin{matrix} \frac{b_{1}}{t^{'} b_{1} + z φ a_{3}} + \frac{a_{3}}{t^{'} a_{3} + y θ b_{1}} - \frac{1}{t^{'}} \\ - (\frac{b_{1}}{t^{'} b_{1} + z φ a_{3}}) \cdot (\frac{(t^{'} - t) a_{3}}{t^{'} a_{3} + y θ b_{1}}) \end{matrix}] \geq 0 \\ {tt}^{'} - yz θ φ \geq 0 \end{matrix}$ and setting of b₁ = a₃ = 0 gives

$\begin{matrix} [\begin{matrix} \frac{a_{1}}{x^{'} a_{1} + y θ b_{3}} + \frac{b_{3}}{x^{'} b_{3} + z φ a_{1}} - \frac{1}{x^{'}} \\ - (\frac{a_{1}}{x^{'} a_{1} + y θ b_{3}}) \cdot (\frac{(x^{'} - x) b_{3}}{x^{'} b_{3} + z φ a_{1}}) \end{matrix}] \geq 0 \\ \land {xx}^{'} - yz θ φ \geq 0 \\ min ({xx}^{'}, {tt}^{'}) \geq yz θ φ \end{matrix}$ (94)

Also setting a₁ = a₃ = 0 and b₁ = b₃ = 0,the inequality (equ32.212) leads to the results $\frac{b_{1}}{t^{'} b_{1} + y θ b_{3}} - \frac{b_{1}}{x^{'} b_{3} + t^{'} b_{1}} \geq 0$ (95) and $\frac{a_{3}}{t^{'} a_{3} + z φ a_{1}} - \frac{a_{3}}{x^{'} a_{1} + t^{'} a_{3}} \geq 0$ (96) respectively. Similarly setting a₃ = b₃ = c = d = 0,the inequality (equ32.21) leads to

$\begin{matrix} [\begin{matrix} \frac{(a_{1} + b_{2})}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{a_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{a_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} \\ - (\frac{a_{1} + b_{2}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})}) \cdot (\frac{(x^{'} - x) a_{2} + (t^{'} - t) b_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}}) \end{matrix}] \\ \geq 0 \end{matrix}$ (97) and $\begin{matrix} [\begin{matrix} \frac{(a_{2} + b_{1})}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})} + \frac{b_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}} \\ - \frac{b_{1}}{x^{'} a_{1} + t^{'} b_{1} + y θ b_{2} + z φ a_{2}} \\ - (\frac{a_{2} + b_{1}}{x^{'} (a_{1} + b_{2}) + t^{'} (a_{2} + b_{1})}) \cdot (\frac{(x^{'} - x) a_{2} + (t^{'} - t) b_{2}}{x^{'} a_{2} + t^{'} b_{2} + y θ b_{1} + z φ a_{1}}) \end{matrix}] \\ \geq 0 \end{matrix}$ (98)

In particular, setting a₁ = a₂ = 0 and a₁ = b₂ = 0 the inequality (equ32.220) leads to the results $\begin{matrix} [\begin{matrix} \frac{b_{1}}{x^{'} b_{2} + t^{'} b_{1}} + \frac{b_{2}}{t^{'} b_{2} + y θ b_{1}} - \frac{b_{1}}{t^{'} b_{1} + y θ b_{2}} \\ - (\frac{b_{1}}{x^{'} b_{2} + t^{'} b_{1}}) \cdot (\frac{(t^{'} - t) b_{2}}{t^{'} b_{2} + y θ b_{1}}) \end{matrix}] \geq 0 \\ {tt}^{'} - (x^{'} - y θ) y θ \geq 0 \end{matrix}$ (99) and the inequality $\frac{1}{t^{'}} - \frac{b_{1}}{t^{'} b_{1} + z φ a_{2}} - (\frac{1}{t^{'}}) \cdot (\frac{(x^{'} - x) a_{2}}{x^{'} a_{2} + y θ b_{1}}) \geq 0$

will lead to same result (equ32.211) and setting b₁ = b₂ = 0 and b₁ = a₂ = 0 the inequality (equ32.219) leads to the results

$\begin{matrix} [\begin{matrix} \frac{a_{1}}{x^{'} a_{1} + t^{'} a_{2}} + \frac{a_{2}}{x^{'} a_{2} + z φ a_{1}} - \frac{a_{1}}{x^{'} a_{1} + z φ a_{2}} \\ - (\frac{a_{1}}{x^{'} a_{1} + t^{'} a_{2}}) \cdot (\frac{(x^{'} - x) a_{2}}{x^{'} a_{2} + z φ a_{1}}) \end{matrix}] \geq 0 \\ {xx}^{'} - (t^{'} - z φ) z φ \geq 0 \end{matrix}$ (100) and the inequality $\frac{1}{x^{'}} - \frac{a_{1}}{x^{'} a_{1} + y θ b_{2}} - (\frac{1}{x^{'}}) \cdot (\frac{(t^{'} - t) b_{2}}{t^{'} b_{2} + z φ a_{1}}) \geq 0$

will ends up with same result as (equ32.27). Now inequalities (equ32.24) and (equ32.222) will be true only if $t^{'} \geq max (y θ, z φ)$ (101) and inequalities (equ32.28),(equ32.217), (equ32.218) and (equ32.221) will be satisfied only if $x^{'} \geq max (y θ, z φ)$ (102)

That is, $\min (x^{'}, t^{'}) \geq max (y θ, z φ)$ (103)

Thus (equ32.25, equ32.26, equ32.27, equ32.29, equ32.210,equ32.211equ32.214,equ32.216 andequ32.225) are the necessary condition for transitivity.

3.3 Min transitive members

Theorem 3.3.1. The family (equ3.03) does not contain any T_M-transitive members.

Proof. The inclusion measure I is T_M-transitive if it fulfill the definition of transitivity (equ1.3). For this purpose, we have to verify that either I (A, B) ≤ I (A, C) or I (B, C) ≤ I (A, C). That is, $\begin{matrix} [1 - \frac{x (| A ∖ B | + | B ∖ A |) + y | A \cap B | + z | {(A \cup B)}^{c} |}{x^{'} (| A ∖ B | + | B ∖ A |) + y θ | A \cap B | + z φ | {(A \cup B)}^{c} |}] \\ - [1 - \frac{x (| A ∖ C | + | C ∖ A |) + y | A \cap C | + z | {(A \cup C)}^{c} |}{x^{'} (| A ∖ C | + | C ∖ A |) + y θ | A \cap C | + z φ | {(A \cup C)}^{c} |}] \geq 0 \end{matrix}$

Using the values of cardinalities given in (equ2.04), the above inequality implies

$\begin{matrix} [1 - \frac{x (a_{1} + b_{2} + a_{2} + b_{1}) + y (b_{3} + c) + z (a_{3} + d)}{x^{'} (a_{1} + b_{2} + a_{2} + b_{1}) + y θ (b_{3} + c) + z φ (a_{3} + d)}] \\ - [1 - \frac{x (a_{1} + b_{3} + a_{3} + b_{1}) + y (b_{2} + c) + z (a_{2} + d)}{x^{'} (a_{1} + b_{3} + a_{3} + b_{1}) + y θ (b_{2} + c) + z φ (a_{2} + d)}] \geq 0 \end{matrix}$ (104)

Setting a₂ = b₂ = c = d = 0 in (equ33.11) and ignoring the factors x′ - x, θ - 1 and φ - 1, the inequality can be simplified as $\frac{a_{1} + b_{1}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} - \frac{1}{x^{'}} \geq 0$ (105)

On simplification, this leads to contradiction as $- \frac{1}{x^{'}} \frac{y θ b_{3} + z φ a_{3}}{x^{'} (a_{1} + b_{1}) + y θ b_{3} + z φ a_{3}} \geq 0$ (106)

Remark 3.3.2. The family (6) does not contain any T_M transitive members.

4 Conclusion

In this paper, we try to tackle the problem arises due the limitation imposed by four parameters and try to discuss the necessary conditions for transitivity of similarity and inclusion measure. This would help to compare objects based not only on positive matching but also on the basis of negative matchings. This will play a significance role medicines, image processing, text mining, etc. where negative match has a vital role. For future directions, similar parametric family can be devised for multiplication of cardinalities.

Footnotes

Acknowledgments

The authors want to thank the referees for their valuable comments that have contributed to an improvement of the paper.

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